Announcement
Course webpage
http://highenergy.phys.ttu.edu/~slee/1408/
Homework.1 – HW due 1/29 (next Tuesday)
Chapter 2
Ch1. 2, 10, 16, 26, 33, 34
Ch.2 4, 6, 12, 16, 22, 32, 40, 41, 49
Motion in One Dimension
Lab
Monday 11:30 am – 12: 50 pm (Sci. 105)
We will consider motion in 1-dimension
i.e. along a straight line
Motion represents a continual change in an object!s position.
Recitation
Monday 3:00 pm – 3:50 pm (Sci. 105)
Quiz.1
1/31 (Next Friday)
Motion & Motion Diagram
Types of Motion
Translational
!! An example is a car traveling on a highway.
Rotational
!! An example is the Earth!s spin on its axis.
Vibrational
!! An example is the back-and-forth movement of a pendulum.
3
Introduction
Making a Motion Diagram
Change and Motion
Several frames from the movie of a car
A stationary ball on the ground.
Same position at all times.
Velocity = 0.
We can superimpose all the frames to create a
“stop-motion” picture that shows the progression
of positions as equal intervals of time pass.
A skateboarder rolling on a sidewalk.
Images are equally spaced.
Velocity = constant.
A sprinter starting the 100 meter dash.
Image spacing grows.
Velocity = increasing.
The equal spacing of the images indicates that
the car has a “constant velocity”
A car stopping for a red light.
Image spacing shrinks.
Velocity = decreasing.
5
If we look at translational motion only, then we can model an extended object
as a particle.
6
Images to Dots to Motion
A stationary ball on the ground.
Same position at all times.
Velocity = 0.
A skateboarder rolling on a sidewalk.
Images are equally spaced.
Velocity = constant.
A sprinter starting the 100 meter dash.
Image spacing grows.
Velocity = increasing.
A car stopping for a red light.
Image spacing shrinks.
Velocity = decreasing.
Actual images can be replaced by ‘dots’ showing successive positions. 7
8
Kinematics, The Math of Motion
Motion in 1-Dimension
"!
Mathematics can be used in Physics because the system
(a ball, a wheel) can be defined very accurately.
"!
To describe motion, we have to learn how to calculate
with vectors. (see Ch.3)
"!
Its purpose is different: Physicists are two lazy to
memorize all different motion diagrams. Math allows one
to describe many different types of motion with a single
equation.
"!
start with the simplest case: motion in one-dimension.
"!
Conveniently we can choose our coordinate system in
such a way that the motion is along the x-axis.
Mathematics makes things easier, that is why it is used in
the first place.
"!
Instead of a vector we then can consider only its
component x (position along x-axis), vx (x-velocity), and ax
(x-acceleration)
"!
9
10
Position-Time Graph
Kinemtics
•! Describes motion while ignoring
the agents that caused the motio
n (e.g. Force)
•! For now, will consider motion in
one dimension
The position-time graph shows the
motion of the particle (car).
The smooth curve is a guess as to what
happened between the data points.
Position
The object!s position is its location with
respect to a chosen reference point.
!! Consider the point to be the origin
of a coordinate system.
Only interested in the car!s
translational motion, so model as a
particle
Section 2.1
Section 2.1
Data Table
Displacement
The table gives the actual data
collected during the motion of the object
(car).
Displacement is defined as the change in position during some time interval.
Positive is defined as being to the right.
!! Represented as !x
!x ! xf - xi
!! SI units are meters (m)
!! !x can be positive or negative
Different than distance !!!
!! Distance is the length of a path followed by a particle.
Section 2.1
Section 2.1
Distance vs. Displacement – An Example
Vectors and Scalars
Assume a player moves from one end
of the court to the other and back.
Vector quantities need both magnitude (size or numerical value) and direction to
completely describe them.
Distance is twice the length of the court
!! Distance is always positive
!! Will use + and – signs to indicate vector directions in this chapter
Scalar quantities are completely described by magnitude only.
Displacement is zero
!! !x = xf – xi = 0 since xf = xi
Section 2.1
Section 2.1
Velocity
Simple Calculations for fun
17
18
Velocity, again!!
Average speed is a measure of how fast (or slowly)
an object is moving. It is a ratio.
Average speed =
Time interval spent traveling
If direction is important, we must use the vector
displacement instead of just distance, and we get
average velocity.
Vector quantities need both magnitude (size or numerical value) and
direction to completely describe them.
"!
Distance traveled
Will use + and – signs to indicate vector directions in this chapter
Average velocity =
Scalar quantities are completely described by magnitude only.
19
Displacement
Time interval for displacement
20
Average Velocity
•! Velocity v is the !rate of change of position"
•! Average velocity vav in the time !t = t2 - t1 is:
x(t2 ) " x(t1 ) #x
vav !
=
t2 " t1
#t
!x
x2
x
trajectory
Vav = slope of line connecting x1 and x2
x1
t1
t2
t
!t
Instantaneous Velocity
The limit of the average velocity as the time interval becomes infinitesimally
short, or as the time interval approaches zero.
!x dx
v x = lim
!t "0
!t
=
dt
The instantaneous velocity indicates what is happening at every point of time.
The instantaneous velocity is the slope
of the line tangent to the x vs. t curve.
!! This would be the green line.
The light blue lines show that as !t gets
smaller, they approach the green line.
The instantaneous speed is the magnitude
of the instantaneous velocity.
Section 2.2
Uniform Motion
Uniform Motion
•! For uniform motion the (average) velocity remains
constant: (i.e. under constant velocity)
"!
"!
The simplest form of motion is uniform motion.
An object’s motion is uniform if its position-versus-time graph is a
straight line. (see Fig)
•! An alternative way to say, for uniform motion, x(t) is a linear
function of t: (see fig in the next slide)
•! The mathematical representation of this situation is the equation.
vx =
!x x f " x i
=
!t
!t
or
x f = x i + v x !t
"!
Fig shows how uniform and non-uniform motion appear in motion
diagrams and (x,t) graphs.
24
Acceleration
Position, time, and velocity are important concepts, and they
might appear to be sufficient. But that is not the case.
Sometimes an object#s velocity changes as it moves.
Acceleration is rate of change of velocity.
Velocity changes if:
the magnitude of the velocity (speed) changes
the direction of the velocity changes
Average acceleration =
a x ,avg
Change of velocity
"v x v xf # v xi
!
=
"t
tf # ti
Time interval
Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as !t
approaches 0.
!v x dv x d 2 x
=
= 2
!t "0 !t
dt
dt
ax = lim
The slope of the velocity-time graph is
the acceleration.
The green line represents the
instantaneous acceleration.
The blue line is the average
acceleration.
Section 2.4
Graphical Comparison
Acceleration and Velocity, Directions
Given the displacement-time graph (a)
When an object!s velocity and acceleration are in the same direction, the object
is speeding up.
When an object!s velocity and acceleration are in the opposite direction, the
object is slowing down.
The velocity-time graph is found by
measuring the slope of the positiontime graph at every instant.
Acceleration and Force
The acceleration-time graph is found by
measuring the slope of the velocity-time
graph at every instant.
Section 2.4
The acceleration of an object is related to the total force exerted on the object.
!! The force is proportional to the acceleration, Fx " ax .
Section 2.4
Constant Velocity
Motion Diagrams
A motion diagram can be formed by imagining the photograph of a moving
object.
Red arrows represent velocity.
Purple arrows represent acceleration.
Images are equally spaced.
The car is moving with constant positive velocity (shown by red arrows
maintaining the same size).
Acceleration equals zero.
Section 2.5
Section 2.5
Acceleration and Velocity, 3
Acceleration and Velocity, 4
Images become farther apart as time increases.
Images become closer together as time increases.
Velocity and acceleration are in the same direction.
Acceleration and velocity are in opposite directions.
Acceleration is uniform (violet arrows maintain the same length).
Acceleration is uniform (violet arrows maintain the same length).
Velocity is increasing (red arrows are getting longer).
Velocity is decreasing (red arrows are getting shorter).
This shows positive acceleration and positive velocity.
Section 2.5
Positive velocity and negative acceleration.
Section 2.5
Kinematic Equations
Kinematic Equations, 1
The kinematic equations can be used with any particle under uniform
acceleration.
For constant ax,
The kinematic equations may be used to solve any problem involving onedimensional motion with a constant acceleration.
You may need to use two of the equations to solve one problem.
v xf = v xi + ax t
Knowing an objects# acceleration is
the key to finding its velocity at later
instants of time.
We can determine an object!s velocity at any time t when we know its
initial velocity and its acceleration
!! Assumes ti = 0 and tf = t
Does not give any information about displacement
Section 2.6
Section 2.6
Kinematic Equations, 2
Kinematic Equations, 3
For constant acceleration,
For constant acceleration,
v x ,avg =
v xi + v xf
2
xf = xi + v x ,avg t = xi +
The average velocity can be expressed as the mean of the initial and final
velocities.
!! This applies only in situations where the acceleration is constant.
Section 2.6
1
(v xi + v fx )t
2
This gives you the position of the particle in terms of time and velocities.
Doesn!t give you the acceleration
Section 2.6
Kinematic Equations, 4
Kinematic Equations, 5
For constant acceleration,
For constant a,
xf = xi + v xi t +
1 2
ax t
2
v xf2 = v xi2 + 2ax (xf ! xi )
Gives final position in terms of velocity and acceleration
Gives final velocity in terms of acceleration and displacement
Doesn!t tell you about final velocity
Does not give any information about the time
Section 2.6
Section 2.6
When a = 0
When the acceleration is zero,
!!vxf = vxi = vx
v xf = v xi + ax t
!!xf = xi + vx t
xf = xi + v xi t +
Section 2.6
1 2
ax t
2
Kinematic Equations – summary
Section 2.6
Graphical Look at Motion: Displacement – Time curve
Graphical Look at Motion: Velocity – Time curve
The slope of the curve is the velocity.
The slope gives the acceleration.
The curved line indicates the velocity is
changing.
The straight line indicates a constant
acceleration.
!! Therefore, there is an acceleration.
Section 2.6
Section 2.6
Graphical Look at Motion: Acceleration – Time curve
The zero slope indicates a constant
acceleration.
Section 2.6